In this paper, we define a new finite-element method for numerically approximating the solution of a partial differential equation in a bulk region coupled to a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation, and apply these to find optimal order error estimates. A numerical experiment is described which demonstrates the order of convergence.

In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ ⊂ R n+1 , is embedded in a polyhedral domain in R n+1 consisting of a union, T h , of (n + 1)-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on T h . Our first method is a sharp interface method, SIF, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γ h , of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes SIF from the method of [42]. The second method, NBM, is a narrow band method in which the region of integration is a narrow band of width O(h). NBM is similar to the method of [13] but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L 2 and H 1 norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence.

We use the evolving surface finite element method to solve a Cahn-Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the paper by deriving error estimates and present various numerical examples.